Circle — Exercise 14.2

If arcs APB and CQD of a circle are congruent, then find the ratio of AB : CD.

Answer

arc APB = arc CQD (Given)

If two arcs are equal then chords are also equal.

∴ AB = CD = x (let).

Ratio = $\dfrac{AB}{CD} = \dfrac{x}{x} = \dfrac{1}{1}$

Hence, AB : CD = 1 : 1.

A and B are points on a circle with center O. C is a point on the circle such that OC bisects ∠AOB, prove that OC bisects the arc AB.

Answer

The figure of the circle is shown below:

Given,

OC bisects ∠AOB.

∴ ∠AOC = ∠BOC

Since, equal arcs subtend equal angles at center.

∴ AC = BC

∴ C is mid-point of AB.

Hence, proved that OC bisects the arc AB.

Prove that the angle subtended at the center of a circle is bisected by the radius passing through the mid-point of arc.

Answer

The figure of the circle is shown below:

Let C be the mid-point of arc AB.

∴ AC = BC.

Since, equal arcs subtend equal angles at center.

∴ ∠AOC = ∠BOC.

Hence, proved that ∠AOC = ∠BOC.

In the adjoining figure, two chords AB and CD of a circle intersect at P. If AB = CD, prove that arc AD = arc CB.

Answer

Given AB = CD

Since, in a circle, equal chords cut off equal arcs.

∴ arc AB = arc CD

Subtracting arc BD from both sides we get,

⇒ arc AB - arc BD = arc CD - arc BD

⇒ arc AD = arc CB.

Hence, proved that arc AD = arc CB.